Connectivity and Hamiltonicity game. Consider the \((1:b)\)  Maker-Breaker connectivity game and Hamiltonicity game played on the edges of some graph \(G\) , where the winning sets are all spanning subgraphs and Hamilton cycles of \(G\) , respectively. The connectivity game was first studied by Lehman [1]}, who showed that Maker as the second player wins the \((1:1)\)  Maker-Breaker connectivity game on any graph \(G\)  if and only if \(G\)  contains two edge-disjoint spanning trees. A natural next step then was to look at the \((1:b)\)  version of the connectivity game played on \(K_n\) . For this, Chvátal and Erdős [2]} proved that the threshold bias is bounded from above by \((1 + o(1))n / \ln n\) , and this bound trivially carries over to the Hamiltonicity game as well. A matching lower bound for the connectivity game was later proven by Gebauer and Szabó [3]}, and for the Hamiltonicity game by Krivelevich [4]}.
